Some potentials for the curvature tensor on three-dimensional manifolds (Q2487322)

The study of the Lanczos potentials for the curvature and Weyl tensors has been pursued by several people and this paper is another contribution to this subject. In such studies, the idea is to express these tensors in terms of another tensor (the potential) and its derivatives, in much the same way as the electromagnetic field is expressed in terms of the vector potential. Such Lanczos potentials have been studied both locally and globally and this paper contributes in both directions. In fact the authors present some ways in which the global extension of such potentials may be topologically obstructed. They then concentrate on the associated ``Ricci-Lanczos'' equation. This equation is obtained from the expression for the curvature tensor (in, terms of its potential) by a contraction, which then gives an expression for the Ricci tensor in terms of a potential. They then examine this equation in 3-dimensions where the Ricci and Riemann tensor are nicely related. They also explore different potential type relations for the Ricci tensor. In this paper the authors prove some results regarding the existence of covariantly constant second order symmetric tensors. These results are similar to results in 4-dimensions proved by the present reviewer many years ago and are more easily proved and understood using holonomy theory (for a summary, see \textit{G. S. Hall}, ``Symmetries and curvature in general relativity'', World Scientific Lecture Notes in Physics 46, River Edge, NJ: World Scientific (2004; Zbl 1054.83001)]). They can, no doubt, be applied to the present situation by using the 3-dimensional classification theory in [\textit{G. S. Hall} and \textit{M. S. Capocci}, J. Math. Phys. 40, No. 3, 1466--1478 (1999; Zbl 0947.83014)].